Fesenko wrote here about "interactions of model theory, arithmetic and algebraic geometry and noncommutative geometry", but I don't remember if he mentiones Iwasawa theory.
A result coming from Y.I. Manin's idea to address the Mordell--Weil problem for cubic surfaces using model theorety are these reconstruction theorems.
BTW, as the André-Oort conjecture is an analogy to the Manin-Mumford conj. , and the later has been treated by model theoretic methods, applies model theory to the former (and common generalizations) too?